Chapter 4 discussion with Lachlan
4.2
False.
The union between \(\{0\} \cup \{p \in \mathcal{P}(\mathbb{F}): deg\ p = m\}\) is not closed under addition. You can add two \(m\) degree polynomials and get something that’s not \(m\) degrees:
\begin{equation} (z^{m} + 1) - z^{m} = 1 \end{equation}
4.3
False.
The union between \(\{0\} \cup \{p \in \mathcal{P}(\mathbb{F}): deg\ p\ even\}\) is not closed also under addition, for the same reason:
\begin{equation} (z^{m} + z^{m-1} + 1) - (z^{m} + 1) = z^{m-1} \end{equation}
One Chapter 5 Exercise
5.A.5
Suppose \(T \in \mathcal{L}(V)\), prove that the intersection of every collection of \(V\) that is invariant under \(T\) is invariant under \(T\)
Let \(U_1 \dots U_{n}\) be invariant subspaces under \(T\).
That is:
\begin{equation} T u_{j} \in U_{j} \end{equation}
We desire that:
\begin{align} Tu \in \bigcap U_{j}\ |\ u \in \bigcap U_{j} \end{align}
WLOG, treat \(u \in \bigcap U_{j}\) as \(u \in U_{j}\). Now, \(Tu \in U_{j}\). This holds \(\forall U_{j}\). Therefore, \(Tu \in \forall U_{j}\). So \(Tu \in \bigcap U_{j}\).
Hence, the intersection of invariant subspaces are invariant as well.